Remark 9.7 In \(\mathbb {R}^n\) , \(n\geqslant 2\) , consider the \(\alpha \) -Riesz kernel \(|x-y|^{\alpha -n}\)  of order \(\alpha \in (0,2]\) , \(\alpha <n\) . Then the concept of inner equilibrium measure as well as its alternative characterizations, provided by formulae (REF ), (REF ), and (), can be generalized to any \(A\subset \mathbb {R}^n\)  that is inner \(\alpha \) -thin at infinity — even if the inner \(\alpha \) -Riesz capacity of \(A\)  is infinite (see [1]} and [2]}). Note that in this generalization, the inner \(\alpha \) -Riesz equilibrium measure \(\gamma _A\)  as well as the admissible measure in (REF ) must be allowed to be of infinite energy.

